Problem: Determine how many solutions exist for the system of equations. ${5x-y = -6}$ ${10x-2y = -12}$
Answer: Convert both equations to slope-intercept form: ${5x-y = -6}$ $5x{-5x} - y = -6{-5x}$ $-y = -6-5x$ $y = 6+5x$ ${y = 5x+6}$ ${10x-2y = -12}$ $10x{-10x} - 2y = -12{-10x}$ $-2y = -12-10x$ $y = 6+5x$ ${y = 5x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+6}$ ${y = 5x+6}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${5x-y = -6}$ is also a solution of ${10x-2y = -12}$, there are infinitely many solutions.